Optimal. Leaf size=44 \[ \frac{(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]
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Rubi [A] time = 0.0173963, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {644, 32} \[ \frac{(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]
Antiderivative was successfully verified.
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Rule 644
Rule 32
Rubi steps
\begin{align*} \int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx &=\left ((d+e x)^{2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}\right ) \int (d+e x)^{-p} \, dx\\ &=\frac{(d+e x)^{1+p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)}\\ \end{align*}
Mathematica [A] time = 0.0160117, size = 31, normalized size = 0.7 \[ \frac{(d+e x)^{p+1} \left (c (d+e x)^2\right )^{-p}}{e-e p} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 44, normalized size = 1. \begin{align*} -{\frac{ \left ( ex+d \right ) ^{1+p}}{e \left ( p-1 \right ) \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15625, size = 39, normalized size = 0.89 \begin{align*} -\frac{e x + d}{{\left (e x + d\right )}^{p} c^{p} e{\left (p - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44774, size = 54, normalized size = 1.23 \begin{align*} -\frac{e x + d}{{\left (e p - e\right )}{\left (e x + d\right )}^{p} c^{p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2373, size = 93, normalized size = 2.11 \begin{align*} -\frac{{\left (x e + d\right )}^{p} x e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \left (c\right ) + 1\right )} +{\left (x e + d\right )}^{p} d e^{\left (-2 \, p \log \left (x e + d\right ) - p \log \left (c\right )\right )}}{p e - e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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